The Properties is the only feature type that does not offer post categories:

How To Add or Edit Categories For Post Types

The reason for this is that the property feature type offers additional filters the members can utilize on the site like Property Status, Property Type, # of Beds, and # of Baths:

All these search filters can be easily edited via the Toolbox >> Form Manager >> Feature - Properties form:

To update the options of the field called All Statuses:

Search for the variable name property_status and make the necessary changes by clicking on Save Changes:

Front-end view after the changes:

Please keep in mind that the wording "All Statuses" is a text label that can be changed under Settings >> Text Labels by conducting a search for "property_all_statuses" variable name:

The process we went over above can be applied to the rest of the search filter options like Property Type (property_type), # of Beds (property_beds), and # of Baths (property_baths).

Back-end Changes:

Front-end View:

Important Note: The format of dropdown options for Property Type is slightly different then others. To add a new dropdown option, please follow the existing format:

So to add a new dropdown option, add a "," after the last option and follow the format as NewOption => NewOption (wording can be different):